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Great retrosnub icosidodecahedron

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Title: Great retrosnub icosidodecahedron  
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Subject: Snub polyhedron, Uniform polyhedra, Great inverted snub icosidodecahedron, Great snub icosidodecahedron, Pentagram
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Great retrosnub icosidodecahedron

Great retrosnub icosidodecahedron
Great retrosnub icosidodecahedron
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Wythoff symbol |3/2 5/3 2
Symmetry group I, [5,3]+, 532
Index references U74, C90, W117
Dual polyhedron Great pentagrammic hexecontahedron
Vertex figure Great retrosnub icosidodecahedron
Bowers acronym Girsid

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It is given a Schläfli symbol s{3/2,5/3}.

Cartesian coordinates

Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),
(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ


β = −ξ/τ+1/τ2−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the smaller positive real root of ξ3−2ξ=−1/τ, namely

\xi=\frac{\left(1+i \sqrt3\right)\left(\frac1{2 \tau}+\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13+ \left(1-i \sqrt3\right)\left(\frac1{2 \tau}-\sqrt{\frac{\tau^{-2}}4-\frac8{27}}\right)^\frac13}2

or approximately 0.3264046. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

See also

External links

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